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Question
By which smallest number must the following number be divided so that the quotient is a perfect cube?
35721
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Solution
On factorising 35721 into prime factors, we get:
\[35721 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 7 \times 7\]
On grouping the factors in triples of equal factors, we get:
\[35721 = \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times 7 \times 7\]
It is evident that the prime factors of 35721 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 35721 is a not perfect cube. However, if the number is divided by (\[7 \times 7 = 49\]), the factors can be grouped into triples of equal factors such that no factor is left over.
Thus, 35721 should be divided by 49 to make it a perfect cube.
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